Commentary on Proposition XI.1¶
Back to Proposition XI.1.
There is no doubt that the proofs of the first three propositions are unsatisfactory owing to the fact that Euclid is not able to make any use of his definition of a plane for the purpose of these proofs, and they really depend upon truths which can only be assumed as axiomatic. The definition of a plane as that surface which lies evenly with the straight lines on itself, whatever its exact meaning may be, is nowhere appealed to as a criterion to show whether a particular surface is or is not a plane. If the meaning of it is what I conjecture in the note on Book I., Def. 7 (Vol. I. p. 171), if, namely, it only tries to express without an appeal to sight what Plato meant by the “middle covering the extremities” (i.e. apparently, in the case of a plane, the fact that a plane looked at edgewise takes the form of a straight line), then it is perhaps possible to connect the definition with a method of generating a plane which has commended itself to many writers as giving a better definition. Thus, if we conceive a straight line in space and a point outside it placed so that, in Plato’s words, the line “covers” the point as we look at them, the line will also “cover” every straight line which passes through the given point and some one point on the given straight line. Hence, if a straight line passing always through a fixed point moves in such a way as to pass successively through every point of a given straight line which does not contain the given point, the moving straight line describes a surface which satisfies the Euclidean definition of a plane as I have interpreted it. But if we adopt the definition of a plane as the surface described by a straight line which, passing through a given point, turns about it in such a way as always to intersect a given straight line not passing through the given point, this definition, though it would help us to prove Eucl. XI. 2, does not give us the fundamental properties of a plane; some postulate is necessary in addition. The same is true even if we take a definition which gives more than is required to determine a plane, the definition known as Simson’s, though it is at least as early as the time of Theon of Smyrna, who says (p. 112, 5) that a plane is a surface such that, if a straight line meet it in two points, the straight line lies wholly in it (ὅλη αὐτῷ ἐφαρμόζεται). This is also called the axiom of the plane. (For some attempts to prove this on the basis of other definitions of a plane see my note on the definition of a plane surface, I. Def. 7.) If this definition or axiom be assumed, Prop. 1 becomes evident, for, as Legendre says, “In accordance with the definition of the plane, when a straight line has two points common with a plane, it lies wholly in the plane.”
Euclid practically assumes the axiom when he says in this proposition “there will be in the plane of reference some straight line continuous with AB.” Clavius tries, unsuccessfully, to deduce this from Euclid’s own definition of a plane; and he seems to admit his failure, because he proceeds to try another tack. Draw, he says, in the plane DE, the straight line CG at right angles to AC, and, again in the plane DE, CF at right angles to CG [I. 11]. Then AC, CF make right angles with CG in the same plane; therefore (I. 14) ACF is a straight line. But this does not really help, because Euclid assumes tacitly, in Book I. as well as Book XI., that a straight line joining two points in a plane lies wholly in that plane.
A curious point in Euclid’s proof is the reason given why two straight lines cannot have a common segment. The argument is precisely that of the “proof” of the same thing given by Proclus on I. 1 (see note on Book I. Post. 2, Vol. I. p. 197) and is of course inconclusive. The fact that two straight lines cannot have a common segment must be taken to be involved in the definition of, and the postulates relating to, the straight line; and the “proof” given here can hardly, I should say, be Euclid’s, though the interpolation, if it be such, must have been made very early.
The proof assumes too that a circle can be described so as to cut BA, BC and BD, or, in other words, it assumes that AD, BC are in one plane; that is, Prop. 1 as we have it really assumes the result of Prop. 2. There is therefore ground for Simson’s alteration of the proof (after the point where BD has been taken in the given plane in a straight line with AB) to the following:
“Let any plane pass through the straight line AD and be turned about it until it pass through the point C.
“And, because the points B, C are in this plane, the straight line BC is in it. [Simson's def.]
Simson, of course, justifies the last inference by reference to his Corollary to I. 11, which, however, as we have seen, is not a valid proof of the assumption, which is really implied in I. Post. 2. An alternative reading, perhaps due to Theon, says, after the words “which is impossible” in the Greek text, “for a straight line does not meet a straight line in more points than one; otherwise the straight lines will coincide.” Simson (who however does not seem to have had the second clause beginning “otherwise” in the text which he used) attacks this alternative reading in a rather confused note chiefly directed against a criticism by Thomas Simpson, without (as it seems to me) sufficient reason. It contains surely a legitimate argument. The supposed straight lines ABC, ABD meet in more than two points, namely in all the points between A and B. But two straight lines cannot have two points common without coinciding altogether; therefore ABC must coincide with ABD.